Let d=1≤d1≤ d2≤···.≤ dn be a non-decreasing sequence of n positive integers, whose sum is even. Let denote the set of graphs with vertex set [n]={1,2,. . .., n} in which the degree of vertex i is di. Let Gn,d be chosen uniformly at random from . Let d=(d1+d2+···.+dn)/n be the average degree. We give a condition on d under which we can show that w.h.p. the chromatic number of is Θ(d/ln d). This condition is satisfied by graphs with exponential tails as well those with power law tails.

We give a way of constructing non-tame automorphisms of a free group of rank 3 in the variety , with p prime.

Biharmonic Lagrangian surfaces of constant mean curvature in complex space forms are classified. A further important point is that new examples of marginally trapped biharmonic Lagrangian surfaces in an indefinite complex Euclidean plane are obtained. This fact suggests that Chen and Ishikawa's classification of marginally trapped biharmonic surfaces [6] is not complete.

In this paper we prove polynomial versions of the Carlson–Simpson theorem and the Graham–Rothschild theorem on parameter sets. To do so we prove a useful extension of the polynomial Hales–Jewett theorem.

Using a variation on the concept of a CS module, we describe exactly when a simple ring is isomorphic to a ring of matrices over a Bézout domain. Our techniques are then applied to characterise simple rings which are right and left Goldie, right and left semihereditary.

Starting from results of Dubé and Mingarelli, Wahlén, and Ehrström, who give conditions that ensure the existence and uniqueness of nonnegative nondecreasing solutions asymptotically constant of the equationwe have been able to reduce their hypotheses in order to obtain the same existence results, at the expense of losing the uniqueness part. The main tool they used is the Banach Fixed Point Theorem, while ours has been the Schauder Fixed Point Theorem together with one version of the Arzelà-Ascoli Theorem.

We give a combinatorial proof of the result of Kahn, Kalai and Linial [16], which states that every balanced boolean function on the n-dimensional boolean cube has a variable with influence of at least . The methods of the proof are then used to recover additional isoperimetric results for the cube, with improved constants.

We also state some conjectures about optimal constants.

In combinatorial optimization, a popular approach to NP-hard problems is the design of approximation algorithms. These algorithms typically run in polynomial time and are guaranteed to produce a solution which is within a known multiplicative factor of optimal. Unfortunately, the known factor is often known to be large in pathological instances. Conventional wisdom holds that, in practice, approximation algorithms will produce solutions closer to optimal than their proven guarantees. In this paper, we use the rigorous-analysis-of-heuristics framework to investigate this conventional wisdom.

We analyse the performance of three related approximation algorithms for the uncapacitated facility location problem (from Jain, Mahdian, Markakis, Saberi and Vazirani (2003) and Mahdian, Ye and Zhang (2002)) when each is applied to an instances created by placing n points uniformly at random in the unit square. We find that, with high probability, these 3 algorithms do not find asymptotically optimal solutions, and, also with high probability, a simple plane partitioning heuristic does find an asymptotically optimal solution.

We study the gaps in the sequence of sums of h pairwise distinct elements of a given sequence in relation to the gaps in the sequence of sums of h not necessarily distinct elements of . We present several results on this topic. One of them gives a negative answer to a question by Burr and Erdős.

In the moduli space of smooth and complex irreducible projective curves of genus g, let be the locus of curves that do not satisfy the Gieseker-Petri theorem. Let be the subvariety of GPg formed by curves C of genus g with a pencil g1d=(V, L∈G1d(C) free of base points for which the Petri map μV:V⊗H0(C,KC⊗L−1)→H0(C,KC) is not injective. For g≥8, we construct in this work a family of irreducible plane curves of genus g with moduli

Let G be a simple graph on n vertices. A conjecture of Bollobás and Eldridge [5] asserts that if then G contains any n vertex graph H with Δ(H) = k. We prove a strengthened version of this conjecture for bipartite, bounded degree H, for sufficiently large n. This is the first result on this conjecture for expander graphs of arbitrary (but bounded) degree. An important tool for the proof is a new version of the Blow-Up Lemma.

In this paper, using a recent critical point theorem of Ricceri, we establish two multiplicity results for the Schrödinger equation of the formwhere are Carathéodory functions, λ and μ two positive parameters.

Daniel Martin B.Sc., M.A., Ph.D., F.R.S.E. was born in Carluke on 16 April 1915, the only child of William and Rose Martin (née Macpherson). The family home in which he was born, Cygnetbank in Clyde Street, had been remodelled and extended by his father, and it was to be Dan's home all his life. His father, who was a carpenter and joiner, had a business based in School Lane, but died as a result of a tragic accident when Dan was only six. Thereafter Dan was brought up single handedly by his mother.

After attending primary school in Carluke from 1920 to 1927, Dan entered the High School of Glasgow. It was during his third year there that he started studying calculus on his own. He became so enthused by the subject that he set his sights on a career teaching mathematics, at university if at all possible. On leaving school in 1932, he embarked on the M.A. honours course in Mathematics and Natural Philosophy at the University of Glasgow. At that time the Mathematics Department was under the leadership of Professor Thomas MacRobert; the honours course in Mathematics consisted mainly of geometry, calculus and analysis, and the combined honours M.A. with Natural Philosophy was the standard course for mathematicians. A highlight of his first session at university was attending a lecture on the origins of the general theory of relativity, given on 20th June 1933 by Albert Einstein. This was the first of a series of occasional lectures on the history of mathematics funded by the George A. Gibson Foundation which had been set up inmemory of the previous head of the Mathematics Department. From then on, relativity was to be one of Dan's great interests, lasting a lifetime; indeed, on holiday in Iona the year before he died, Dan's choice of holiday reading included three of Einstein's papers.

The weighted energy theory for Navier-Stokes equations in 2D strips is developed. Based on this theory, the existence of a solution in the uniformly local phase space (without any spatial decaying assumptions), its uniqueness and the existence of a global attractor are verified. In particular, this phase space contains the 2D Poiseuille flows.

Let X be a Banach function space over a nonatomic probability space. We investigate certain martingale inequalities in X that generalize those studied by A. M. Garsia. We give necessary and sufficient conditions on X for the inequalities to be valid.

Let , let be the quantum function algebra – over – associated to G, and let be the specialisation of the latter at a root of unity ϵ, whose order ℓ is odd. There is a quantum Frobenius morphism that embeds the function algebra of G, in as a central Hopf subalgebra, so that is a module over . When , it is known by [3], [4] that (the complexification of) such a module is free, with rank ℓdim(G). In this note we prove a PBW-like theorem for , and we show that – when G is Matn or GLn – it yields explicit bases of over . As a direct application, we prove that and are free Frobenius extensions over and , thus extending some results of [5].

Let S be a finite set of integers. We consider a problem of finding D(S), the minimum size of a set A, such that S⊆ A−A. We give a characterization for ‘extremal’ sets and prove lower and upper bounds on D(S) in terms of additive properties of S.